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A modular Poincaré-Wirtinger type inequality on Lipschitz domains for Sobolev spaces with variable exponents

Elisa Davoli, Giovanni Di Fratta, Alberto Fiorenza, Leon Happ

Abstract

In the context of Sobolev spaces with variable exponents, Poincaré--Wirtinger inequalities are possible as soon as Luxemburg norms are considered. On the other hand, modular versions of the inequalities in the expected form \begin{equation*} \int_Ω\left|f(x)-\langle f\rangle_Ω\right|^{p(x)} \ {\mathrm{d} x} \leqslant C \int_Ω|\nabla f(x)|^{p(x)}{\mathrm{d} x}, \end{equation*} are known to be \emph{false}. As a result, all available modular versions of the Poincaré- Wirtinger inequality in the variable-exponent setting always contain extra terms that do not disappear in the constant exponent case, preventing such inequalities from reducing to the classical ones in the constant exponent setting. Our contribution is threefold. First, we establish that a modular Poincaré--Wirtinger inequality particularizing to the classical one in the constant exponent case is indeed conceivable. We show that if $Ω\subset \mathbb{R}^n$ is a bounded Lipschitz domain, and if $p\in L^\infty(Ω)$, $p \geq 1$, then for every $f\in C^\infty(\barΩ)$ the following generalized Poincaré--Wirtinger inequality holds \begin{equation*} \int_Ω\left|f(x)-\langle f\rangle_Ω\right|^{p(x)} \ {\mathrm{d} x} \leq C \int_Ω\int_Ω\frac{|\nabla f(z)|^{p(x)}}{|z-x|^{n-1}}\ {\mathrm{d} z}{\mathrm{d} x}, \end{equation*} where $\langle f\rangle_Ω$ denotes the mean of $f$ over $Ω$, and $C>0$ is a positive constant depending only on $Ω$ and $\|p\|_{L^\infty(Ω)}$. Second, our argument is concise and constructive and does not rely on compactness results. Third, we additionally provide geometric information on the best Poincaré--Wirtinger constant on Lipschitz domains.

A modular Poincaré-Wirtinger type inequality on Lipschitz domains for Sobolev spaces with variable exponents

Abstract

In the context of Sobolev spaces with variable exponents, Poincaré--Wirtinger inequalities are possible as soon as Luxemburg norms are considered. On the other hand, modular versions of the inequalities in the expected form \begin{equation*} \int_Ω\left|f(x)-\langle f\rangle_Ω\right|^{p(x)} \ {\mathrm{d} x} \leqslant C \int_Ω|\nabla f(x)|^{p(x)}{\mathrm{d} x}, \end{equation*} are known to be \emph{false}. As a result, all available modular versions of the Poincaré- Wirtinger inequality in the variable-exponent setting always contain extra terms that do not disappear in the constant exponent case, preventing such inequalities from reducing to the classical ones in the constant exponent setting. Our contribution is threefold. First, we establish that a modular Poincaré--Wirtinger inequality particularizing to the classical one in the constant exponent case is indeed conceivable. We show that if is a bounded Lipschitz domain, and if , , then for every the following generalized Poincaré--Wirtinger inequality holds \begin{equation*} \int_Ω\left|f(x)-\langle f\rangle_Ω\right|^{p(x)} \ {\mathrm{d} x} \leq C \int_Ω\int_Ω\frac{|\nabla f(z)|^{p(x)}}{|z-x|^{n-1}}\ {\mathrm{d} z}{\mathrm{d} x}, \end{equation*} where denotes the mean of over , and is a positive constant depending only on and . Second, our argument is concise and constructive and does not rely on compactness results. Third, we additionally provide geometric information on the best Poincaré--Wirtinger constant on Lipschitz domains.
Paper Structure (7 sections, 12 theorems, 82 equations, 1 figure)

This paper contains 7 sections, 12 theorems, 82 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary definitions and results
  3. Proof of Theorem \ref{['thm-pw']}
  4. Step 1.
  5. Step 2.
  6. Step 3.
  7. Auxiliary Results

Key Result

Theorem 1.1

Let $\Omega\subset {\mathbb{R}^{n}}$ be a bounded Lipschitz domain (open and connected). Let further the variable exponent be in $L^\infty(\Omega)$. Then for all $f\in C^\infty(\overline{\Omega})$ there holds for some constant $C>0$ depending only on $\Omega$ and $p_+:=\|p\|_{L^\infty(\Omega)}$ (see (eq-final-exp)).

Figures (1)

  • Figure 1: Two counterexamples to the sufficiency of (\ref{['eq-ls']}). In the left picture $D$ is the slit disk and $S$ is just a point. In the right picture $D$ is constructed by removing a whole quarter from the unit disk and $S$ is a closed square. In both cases there exist rays starting at a point in $S$ that have infinite intersections with the boundary of $D$.

Theorems & Definitions (28)

  • Theorem 1.1: Poincaré--Wirtinger-type inequality on Lipschitz domains
  • Corollary 1.2
  • Example 2.1
  • Definition 2.2: Starshaped domains with respect to a subset
  • Definition 2.3
  • Proposition 2.4
  • Lemma 3.1
  • Remark 3.2
  • proof : Proof of Lemma \ref{['lem-morrey-ss']}
  • Corollary 3.3
  • ...and 18 more